Inelastic X-Ray Scattering and X-Ray Powder Diffraction

Inquiries regarding the use of the book should be directed to the rights and permissions department of INTECHOPEN LIMITED (permissions@intechopen.com). If the perturbation induced in the target sample is weak, the dependence of the intensity of the scattered beam on the energy and.

Scattering

Introduction

A suitable choice of the exchanged wave vector amplitude Q = |Q| enωallows to tune the probe to dynamic events occurring at different scales. Indeed, the investigation of the collective dynamics in disordered systems imposes access to energy transfer E = ℏω as low as a few meV.

Generalities on an inelastic scattering measurement

Rather, more practical strengths are the significantly higher photon fluxes hitting the sample and the smaller transverse size of the beam. In summary, the optimal sample thickness should match the absorption wavelength of the sample at the energy of the incident beam.

The interaction between impinging electromagnetic field and target electrons

The integration of both members of the above equation leads to the conclusion that the intensity transmitted through the sample experiences an exponential decay. The integration of both members of the above equation leads to the conclusion that the intensity transmitted through the sample experiences an exponential decay.

Counting the photon states

At this stage, the derivation of the double differential cross section requires one to treat the final density of states d2n=dΩdEf analytically, as discussed in the next paragraph. At this stage, the derivation of the double differential cross section requires one to treat the final density of states d2n=dΩdEf analytically, as discussed in the next paragraph.

Introducing a key stochastic variable: the microscopic density fluctuation

The physical meaning of this variable will be discussed in more detail in the next section. In summary, as a result of all manipulations mentioned above, the double differential cross section in Eq.

The double differential cross-section and the dynamic structure factor

In the reciprocal space one deals with the Fourier transform of the microscopic density, namely. By definition, the spectrum of such a variable is the Fourier transform of the autocorrelation function.

An estimate of the count rate

• The signal measured by a real instrument
• A practical example: a comparison between an IXS and an INS measurement

This expression of the cross section above has been derived assuming a target sample composed of N identical atoms and within Born-Oppenheimer. As a result of the previous treatment, it was shown that the cross section is proportional to S Q,ð ωÞ.

Conclusion

However, this performance imposes a general shrinkage of the excited frequency range, which does not include the high-frequency shoulder in the IXS spectrum. In the last decade, a large amount of inelastic neutron and X-ray scattering measurements focused on the study of the collective atomic dynamics in disordered systems [1–5].

An example: searching for differences

The purpose of this chapter is to illustrate how Bayesian inference can be used in X-ray and neutron scattering applications. This Bayesian method has already been successfully applied in a series of Brillouin inelastic neutron scattering works [12], as well as inelastic X-ray scattering works [13, 14] and, very recently, in the description of the time correlation function decay in the time domain as measured by spin echo neutron scattering [15, 16].

Bayesian inference

• The Bayes theorem
• The prior distribution
• The likelihood function
• The posterior distribution and its normalizing constant
• The Occam’s razor principle
• Bayesian computation of model parameters

The probability function is the total probability of the observed data, which depends on the adopted model and the values ​​of its parameters. Conditionally with a certain value of the parameter vector Θ (which may also include the variance σ2i of the error term) we can calculate S Q, Eð iÞand thus P yjΘð Þ. 3.4 Posterior distribution and its normalization constant.

The Bayesian approach in neutron and X-ray scattering spectroscopy

• Neutron and X-ray Brillouin scattering
• Bayesian inference in the time domain

The entire vector of parameters for the model in Eq. ð Þ, so the dimension of the parameter vector depends on the number of inelastic modes, k. From a Bayesian point of view, appropriate prior values ​​must be chosen for each component of Θ. In practice, at the highest Q investigated (16 nm�1), the oscillation mode becomes so strongly damped that it can be fitted equally well with either two distinct DHO peaks or a (broader) one in the middle of the two. From the output of the MCMC-RJ algorithm, the values ​​of the discrete posterior distribution function k are in Table 1.

Posterior probability for the number of k modes at different values ​​of the momentum transfer Q in an NSE experiment performed on polymer solution of polyethylene glycol with a molecular weight of 2000D PEG2000ð Þ in D2O. Values ​​of the quantity s2 as defined in Eq. 22) calculated for the different values ​​of k and taking into account the means of the model parameter posterior distribution.

Figure 4 shows the trace plots of three DHO-mode frequencies ð Ω 1,2,3 Þ the algo- algo-rithm found, fitting a spectrum relative to IXS data on pure water recently mea-sured (data not published) at room temperature and at a wave vector transfer Q ¼ 3nm �1

Van Hove function

• Definition
• Evolution with time

Another step of the Fourier transformation, this time from moment space to real space, leads to the van Hove function [12]. Then the eigenterm of the van Hove function should be in the diffusion regime. Therefore, the self-diffusion coefficient, Di, can be determined from the eigenpart of the van Hove function.

The early prediction about the distinct part of the van Hove function was that it could be expressed by the convolution of the PDF of the self part (Eq. But de Gennes noticed that the QES becomes anomalously narrow near the first peak in S(Q) [17].

Local dynamics of water and aqueous solution of salt

• Van Hove function of water
• Self-diffusion
• Van Hove function of salty water

The part of the van Hove function near r = 0 describes the autocorrelation, Gs(r, t). circles) experimental data and (dashed line) result of fitting with Eq. However, the diffusivity values ​​determined from Eq. 8) differ from the values ​​obtained by other methods [24]. The range between the dashed lines (R10 and R10 0) was used to calculate the first neighbor area A(t). The part of the van Hove function near r = 0 describes the autocorrelation, Gs(r, t). circles) experimental data and (dashed line) result of fitting with Eq. Inelastic scattering of X-rays and applications of X-ray powder diffraction.

As shown in Figure 5, the height of the first peak of the van Hove function is reduced by salt. The time dependence of the area of ​​the first peak above G(r, t) = 1, shown in Figure 6, shows that the addition of salt increases the slowly decaying component.

Figure 2 shows the S(Q, E) of water at room temperature, determined by the IXS experiment at the beam line XL35 of the SPring-8 facility [14]

Limitations of the method

Furthermore, it is possible to decompose the van Hove function into the water-water correlation, Gw-w, and the water-salt correlation, Gw-s. Assuming that Gw-wis is the same as for pure water, we can determine Gw-s from Eq. The decay of the area of ​​the subpeak at 3.2 Å, corresponding to the Cl–O distance, is also the same for all concentrations as shown in Figure 8 , proving that the effect of salt on the dynamics is local.

For the determination of the van Hove function, the current setup of IXS is ideally suited for the study of local dynamics in the time scale of 0.1–2 ps and length scale up to 5 Å. To exceed this limit, we either have to resort to neutron scattering which offers better energy resolution or develop the method of X-ray photon correlation spectroscopy (XPCS) with free electron X-ray laser [26].

Concluding remarks

Mössbauer gamma rays are used as probe beams in unique quasi-elastic scattering spectroscopy with neV energy resolution. In these cases, the ratio of gamma ray energy to natural energy width reaches Γ0=E01013, indicating that Mössbauer gamma rays exhibit very high monochromaticity. The neV energy resolution of gamma rays from 57Fe nuclei allows the dynamics to be measured on time scales of about 100 ns.

In this chapter we consider Mössbauer gamma rays from 57Fe nuclei because the gamma ray is most often used for QEGS spectroscopy. Quasi-elastic scattering spectroscopy using gamma rays from 57Fe covers a unique time and length scale range.

Quasi-elastic scattering spectroscopy using Mössbauer gamma rays In this section, we introduce the quasi-elastic scattering technique using

• Introduction to quasi-elastic scattering
• Energy-domain spectroscopy of QEGS
• Time-domain measurement of QEGS

In quasi-elastic scattering processes, a non-V-energy propagation of gamma-ray energy is observed, as shown in Fig. 2. 57Fe, which allows the measurement of the time spectrum of delayed gamma-rays with high precision. We can see the decay of gamma ray intensity on the excitation time scale.

Next, we considered the time spectrum of gamma rays received by the detector for the QEGS case. The broadening of gamma rays with an energy widthΓ reflects the dynamics in a sample.

Figure 3a shows the common experimental design of the technique [8, 9]. In the setup, monochromatic Mössbauer gamma rays impinge on the sample

Application results of SR-based QEGS using TDI

• Microscopic dynamics in glass formers
• Results on o-terphenyl
• Results on polybutadiene
• Results on polybutadiene with nano-silica

The branching temperature of the JG-β process from the α Tαβ process is often seen close to the dynamic transition temperature Tc. In other words, this transition corresponds to the switching of the temperature dependence from the VFT law to the Arrhenius law. This discrepancy must be due to strong relaxation cooperativity at distances of approx.

In addition, the effect of the particle size on the microscopic dynamics has not been elucidated. Temperature dependence of the average relaxation times obtained for pure PB, PB-silica20 and PB-silica100 at q = 14 nm�1.

Figure 6 depicts the temperature dependence of h i. At q = 14 nm τ �1 , the temperature dependence obeys the VFT law, as suggested by the comparison with best-fitting curve obtained by the least-squares method

Conclusions and perspectives

The results obtained therefore indicate that the polymerα relaxation dynamics were limited by contact with the surfaces of the nanoparticles and became even more limited as the surface area increased. The combination of these new X-ray (and gamma-ray) based techniques significantly extends the time scales of the measurements (e.g. from femtoseconds to microseconds). This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ . by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Influence of cooperative dynamics on local β-relaxation during the development of the dynamic glass transition in poly(n-alkyl methacrylates). Connections of activated hop processes with the breakdown of the Stokes-Einstein relation and with aspects of dynamical heterogeneities.

Ray Powder Diffraction

• Peak refinement and quantitative phase analysis: the Rietveld method X-ray profile fitting provides important crystallographic information from the
• Case 1: abrasive blasting in steel surfaces—addressing contamination by X-ray diffraction quantitative analysis
• Surface roughness corrections
• Experimental parameters
• X-ray analysis results
• Case 2: ferrite/austenite ( α / γ ) ratio in duplex steels and the occurrence of sigma phase: quantification of unbalanced phase
• Heat treatments for different amounts of phase formation
• X-ray analysis results
• Conclusions
• Materials and methods
• Results and discussions
• X-ray diffraction analysis and microstructural study
• Physical properties of the TMCs
• Conclusions

A general overview of the Rietveld profile fitting and quantitative phase analysis is provided in the following sections. An interesting route to promote the strengthening of the matrix is ​​the "in situ" formation of secondary phases. For that reason, the pores are only observed in the middle of the mentioned agglomerations (see Figure 5).

The highest temperature (1200°C) played a major role in the formation of TiB, independent of the operating temperature. Microstructural study of these TMCs confirmed the visual existence of the in situ TiBx phases. This is consistent with the intensity of the peaks of this phase in the TMC patterns (Figure 10).

Analysis of the influence of input materials and processing conditions on the properties of W/Cu alloys.

Table 1 presents the GOF values for each calculation. The calculated values lied between 1 and 1.5, which is an indication of a satisfactory fitting

Scattering and X-Ray Powder Diffraction